Hi,

when I calculate (11**19)%23 it gives me 15 but when I try to do the same operation manually it give a totally different result which is 1395.

>>> 11**19
61159090448414546291
>>> (11**19)%23
15
>>> a=11**19
>>> a
61159090448414546291
>>> b=23
>>> c=int(a/b)
>>> c
2659090889061501952
>>> d=b*c
>>> d
61159090448414544896
>>> e=a-d
>>> e
1395

You should use a//b instead of int(a/b).

>>> 11**19 - 11**19//23*23
15
>>> 11**19 - int(11**19/23)*23
1395

That is because of floating point rounding.

When you calculate a/23, the result is the approximate float 2.659090889061502e+18 instead of the exact integer result 2659090889061502012. Converting to an int gives you a result which is too small:

py> a = 11**19
py> a - (a//23)*23  # calculate modulus with no rounding error
15
py> a - int(a/23)*23  # introduces rounding error
1395
py> a/23
2.659090889061502e+18
py> int(a/23)
2659090889061501952
py> int(a/23) - a//23
-60


By the way, are you aware of the third argument to pow()?

py> pow(11, 19, 23)
15